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| author | pommicket <leonardomtenenbaum@gmail.com> | 2016-07-01 17:43:58 -0400 |
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| committer | pommicket <leonardomtenenbaum@gmail.com> | 2016-07-01 17:43:58 -0400 |
| commit | be06edbcfd1213ea1658d36f8934493a92205aac (patch) | |
| tree | 28dca6130833e3a770611961bc6559c08bed4161 /mandelbrot_explanation.html | |
Created pommicket.github.io
Diffstat (limited to 'mandelbrot_explanation.html')
| -rw-r--r-- | mandelbrot_explanation.html | 72 |
1 files changed, 72 insertions, 0 deletions
diff --git a/mandelbrot_explanation.html b/mandelbrot_explanation.html new file mode 100644 index 0000000..3a59639 --- /dev/null +++ b/mandelbrot_explanation.html @@ -0,0 +1,72 @@ +<html> +<head> +<script src="js/latexit.js"></script> +<link rel="stylesheet" href="css/bootstrap.min.css"> +<link rel="stylesheet" href="css/style.css"> + +<title>Mandelbrot Set Explanation</title> +</head> + +<body> + +<h2>Explanation of the Mandelbrot Set</h2> +<div id="header_links_div"></div> +<script src="js/header_links.js"></script> +<hr> + +Consider the function +<div lang="latex"> +f_c(z) = z^2+c\\ +</div><br> +Where z and c are complex numbers. Complex numbers are numbers in the form of +<div lang="latex"> +\\ +ai+b\\ +$Where $i=\sqrt{-1} +</div> +<br> +Now let's check if 0.5 is in the Mandelbrot Set. To do so, start at 0 +<div lang="latex"> +\\ +f_{0.5}(0) = 0^2 + 0.5 = 0.5\\ +f_{0.5}(0.5) = 0.5^2 + 0.5 = 0.75\\ +f_{0.5}(0.75) = 1.0625\\ +f_{0.5}(1.0625) = 1.62890625\\ +f_{0.5}(1.62890625) = 3.15333557\\ +</div> +<br> +It can be proven that if the function passes 2, it will go to infinity if you continually apply the function. +Since this function has passed 2, 0.5 is not in the Mandelbrot Set. Compare this to 0.25. + +<div lang="latex"> +\\ +f_{0.25}(0) = 0^2+0.25 = 0.25\\ +f_{0.25}(0.25) = 0.3125\\ +f_{0.25}(0.3125) = 0.34765625\\ +f_{0.25}(0.34765625) = 0.370864868\\ +f_{0.25}(0.370864868) = 0.38754075\\ +f_{0.25}(0.38754075) = 0.400187833\\ +f_{0.25}(0.400187833) = 0.410150302\\ +f_{0.25}(0.410150) = 0.418223\\ +f_{0.25}(0.418223) = 0.424911\\ +f_{0.25}(0.424911) = 0.430549\\ +f_{0.25}(0.430549) = 0.435373\\ +f_{0.25}(0.435373) = 0.439549\\ +f_{0.25}(0.439549) = 0.443204\\ +f_{0.25}(0.443204) = 0.446429\\ +f_{0.25}(0.446429) = 0.449299\\ +</div> +<br> +This will never pass 2, so 0.25 is in the Mandelbrot Set. +<br> +This process can also be done to complex numbers.<br> +<br> +<div lang="latex"> +M(x) =$ the number of iterations required for $f_x$ to pass 2.$ +</div> + +The website is just a 2d plot of M(x). + +</body> + +</html> |